A Graph Neural Network (GNN) encoder computes \(d\) -dimensional representations for each node in the input TSP graph.
At each layer, nodes gather features from their neighbors to represent local graph structure via recursive message passing [1]}.
Stacking \(L\)  layers allows the network to build representations from the \(L\) -hop neighborhood of each node.
Let \(h_i^{\ell }\)  and \(e_{ij}^{\ell }\)  denote respectively the node and edge feature at layer \(\ell \)  associated with node \(i\)  and edge \(ij\) .
We define the feature at the next layer via an anisotropic message passing scheme using an edge gating mechanism [2]}, [3]}:
\(h_i^{\ell +1} &=& h_i^{\ell } + \text{ReLU} \Big ( \textsc {Norm} \Big ( U^\ell h_i^{\ell } + \textsc {Aggr}_{j \in \mathcal {N}_i} \Big ( \sigma (e_{ij}^{\ell }) \odot V^\ell h_j^{\ell } \Big ) \Big )\Big ) \ ,  \\e_{ij}^{\ell +1} &=& e^\ell _{ij} + \text{ReLU}\Big ( \textsc {Norm} \Big ( A^\ell e_{ij}^{\ell } + B^\ell h^{\ell }_i + C^\ell h^{\ell }_j \Big )\Big ) , \) 
